Question

The probability distribution of the random variable X is given by

X	0	1	2	3
P(X)	0.2	k	2k	2k

Find the variance of the random variable \(X\).

• A. \(\frac{764}{625}\)
• B. \(\frac{1}{625}\)
• C. 1
• D. \(\frac{108}{25}\)

Answer 1

The Correct Option is A

Step 1: Conceptual Understanding:

This question assesses the ability to calculate the variance of a discrete random variable from its probability distribution. The process involves first determining the unknown constant 'k' by ensuring the sum of all probabilities equals 1. Subsequently, the mean ( \( E[X] \) ) and the expected value of the squared variable ( \( E[X^2] \) ) are computed. The variance is then derived using the standard formula.

Step 2: Essential Formulas and Methodology:

1. Probability Summation: \( \sum P(X_i) = 1 \).
2. Mean (Expected Value): \( E[X] = \mu = \sum X_i P(X_i) \).
3. Expected Value of \( X^2 \): \( E[X^2] = \sum X_i^2 P(X_i) \).
4. Variance Formula: \( \text{Var}(X) = \sigma^2 = E[X^2] - (E[X])^2 \).

Step 3: Detailed Calculation:

Determination of k:
The sum of probabilities must equal 1. \[ 0.2 + k + 2k + 2k = 1 \] \[ 0.2 + 5k = 1 \] \[ 5k = 0.8 \implies k = \frac{0.8}{5} = 0.16 \] The completed probability distribution table is as follows:

X	0	1	2	3
P(X)	0.2	0.16	0.32	0.32

Calculation of Mean \( E[X] \):
\[ E[X] = (0 \times 0.2) + (1 \times 0.16) + (2 \times 0.32) + (3 \times 0.32) \] \[ E[X] = 0 + 0.16 + 0.64 + 0.96 = 1.76 \]

Calculation of \( E[X^2] \):
\[ E[X^2] = (0^2 \times 0.2) + (1^2 \times 0.16) + (2^2 \times 0.32) + (3^2 \times 0.32) \] \[ E[X^2] = (0 \times 0.2) + (1 \times 0.16) + (4 \times 0.32) + (9 \times 0.32) \] \[ E[X^2] = 0 + 0.16 + 1.28 + 2.88 = 4.32 \]

Calculation of Variance \( \text{Var}(X) \):
\[ \text{Var}(X) = E[X^2] - (E[X])^2 \] \[ \text{Var}(X) = 4.32 - (1.76)^2 \] \[ \text{Var}(X) = 4.32 - 3.0976 = 1.2224 \]

Conversion to Fraction:
Comparing with the provided options.
Option (A): \( \frac{764}{625} \).
The decimal value of this fraction is: \( 764 \div 625 = 1.2224 \).
This value matches the calculated variance.

Step 4: Final Result:

The variance of the random variable \( X \) is 1.2224, which is equivalent to \( \frac{764}{625} \).